We present a scheme of studying the geometry of high-dimensional data to discover patterns in it, using minimal parametric distributional assumptions. Our approach is to define multivariate quantiles and extremes, and develop a method of center-outward partial ordering of observations. We formulate methods for quantifying relationships among observed variables, thus generalizing the notions of regression and principal components. We devise geometric algorithms for detection of outliers in high dimensions, classification and supervised learning. Examples on the use the proposed methods will be provided. This is joint work with several students.